Let be a bijective function and let be another function satisfying

MATHEMATICS · Subjective · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Let

f:X\to X

be a bijective function and let

g:X\to X

be another function satisfying

g\circ f = I_X

. (i) State the definition of the inverse of a function. (ii) Prove that

g

is the unique inverse of

f

, that is, show that

g = f^{-1}

and hence

f\circ g = I_X

. (iii) Give one explicit example of two sets

A

and

B

with

f:A\to B

and

g:B\to A

such that

g\circ f = I_A

but

f\circ g \ne I_B

. Explain why the failure of the second composition prevents

g

from being the inverse of

f

in this case.